Number System
Category : 6th Class
NUMBER SYSTEM
FUNDAMENTALS
0, 1, 2, 3,4,5,6,7,8,9
All the other numbers are written using 10 symbols.
Example: 2, 3, 4, 5 etc.
Example: 243, 67842, 546380, etc. are numerals.
\[\Rightarrow \] Let us see the chart of Indian system and understand about Indian system.
Periods |
Crores |
Lakhs |
Thousands |
Ones |
|||||
Places |
Ten Crores 100000000 |
Crores 10000000 |
Ten Lakhs 1000000 |
Lakhs 100000 |
Ten Thousands 10000 |
Thousand 1000 |
Hundreds 100 |
Tens 10 |
Units 1 |
Example: 67842678 = Six Crore Seventy Eight Lakhs Forty Two Thousand Six Hundred Seventy Eight
\[\Rightarrow \] Let us see the chart of International system and understand about International system.
Periods |
Millions |
Thousands |
Ones |
||||||
Places |
Hundred Millions 100000000 |
Ten Millions 10000000 |
Millions 1000000 |
Hundred Thousand 100000 |
Ten Thousands 10000 |
Thousand 1000 |
Hundreds 100 |
Tens 10 |
Units 1 |
Example: 1234567 = One Million Two Hundred Thirty Four Thousand Five Hundred Sixty Seven.
A comma is inserted after each period in Hindu-Arabic system as well as British system.
Example:
1. Six Crore Seventy Eight Lakhs Forty Two Thousand Six Hundred Seventy Eight.
6, 78, 42, 678
2. One Million Two Hundred Thirty Four Thousand Five Hundred Sixty Seven.
1, 234, 567
Place value and Face value
Example: Eighty two thousand four hundred seventy two, that is 82472.
The face value of 2 is 2. Similarly the face value of 7 is 7 and 4, 2 and 8 are 4, 2 and 8 respectively.
PLACE VALUE:
2 has the place value \[2\times 1=2\] (ones place)
7 has the place value \[7\times 10=70\] (Tens place)
4 has the place value \[4\times 100=400\] (Hundreds place)
2 has the place value \[2\times 1000=2000\] (Thousands place)
8 has the place value \[8\times 10000=80000\] (Ten Thousands place)
Note: Place value of a digit = (face value) \[\times \] (value of place).
ROMAN NUMBER
\[\Rightarrow \] Roman symbols and their corresponding Indo-Arabic Numerals.
Roman numerals |
I |
V |
X |
L |
C |
D |
M |
Indo-Arabic Numerals |
1 |
5 |
10 |
50 |
100 |
500 |
1000 |
Roman Numerals Fundamental Rules
Example: \[III=1+1+1=3\]
\[XXX=10+10+10=30\]
\[CC=100+100=200\]
\[III=1000+1000+1000=3000.\]
It may "be note that no numerals can be repeated more than 3 times. Repetition is allowed only for symbols I, X and C.
Estimation
Examples:
1. 54 = 50 (to the nearest tens)
2. 543 = 500 (to the nearest hundreds)
3. 5678 = 6000 (to the nearest thousand)
INTEGERS
\[\therefore \]\[Z=\{---,-4,-3,-2,-1,0,1,2,3,4,----\}\]
NUMBER LINE
Example: \[-5<\text{ }5,-7<\text{ }1,-9<\text{ }3\]
Note: A number line drawn vertically, helps to know the heights above the and below the sea levels which are denoted by positive and negative signs respectively.
Absolute value of an integer
Example: \[|+5|=5,\,\,|-5|=5,\,\,|0|=0\]
If x represents an integer then,
\[|x|=x,\] if \[x\] is positive or zero
\[=-x\] if \[x\] is negative.
Fundamental Operations on Integers
Addition of two Integers:
Example 1: \[(-2)+(-1)=-3.\]
We observe from the number line that \[(-2)+(-1)\]means '2' units to the left of '0' and '1' units to the left of \[(-1)\] gives 3 units to left of zero. i.e.\[(-2)+(-1)=-3.\]
Example 2: \[(-2)+(1)=-1.\]
We observe from the number line that \[\{(-2)+1\}\]means '2' units to the left of zero and '1' unit right of \['-2'\] gives '1' unit left of zero. i.e. \[(-2)+1=-1.\]
Subtraction of Integers
Example 1: \[7-2=5\]here, 5 gets the sign of 7 and on number line it is represented as follows.
i.e. \[7-2\]means 2 unit to left of 7.
Example 2: \[-7-(-2)=-7+2=-5\]
ie. \[-7-(-2)\] follows the rule of sign and becomes \[-7+2\] which is equal to 2 unit to the right of \[-7\] on number line.
Example 3: \[-7-(2)=-9\]
i.e., \[-7-2\]means 2 units to the left of\[-7\]and gives\[-9\].
Things to Remember
\[-\times -=+\] |
\[-\,2\times -2=+\,4\] |
Sum of consecutive natural numbers \[=\frac{n\,(n+1)}{2}\] e.g, 1 + 2 + 3 + 4 + ???. + 50 |
\[-\times +=-\] |
\[-\,3\times 1=-\,3\] |
|
\[+\times -=-\] |
\[4\times -2=-\,8\]. |
|
\[+\times +=+\] |
\[2\times 6=12\] |
Solution: \[\frac{50\,(51)}{2}=25\times 51=1275\] |
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